On the isometric minimal immersion into a euclidean space

We study the volume growth of the geodesic balls of a minimal submanifold in a Euclidean space. A necessary condition for the isometric minimal immersion into a Euclidean space is obtained. A classification of non-positively curved minimal hypersurfaces in a Euclidean space is given. This work is partially supported by the National Science Foundation of China     

高维欧氏空间中向量的外积

指出了对高维欧式空间中向量外积定义的不足,从几何空间中向量外积的几何描述入手,经过简洁的证明推导,重新提出了高维欧式空间中向量外积的定义,并得出了若干相关结论.     

Projections onto convex sets on the sphere

In this paper some concepts of convex analysis are extended in an intrinsic way from the Euclidean space to the sphere. In particular, relations between convex sets in the sphere and pointed convex cones are presented. Several characterizations of the usual projection onto a Euclidean convex set are extended to the sphere and an extension of Moreau’s theorem for projection onto a pointed convex cone is exhibited.     

Fréchet Embeddings of Negative Type Metrics

We show that every n-point metric of negative type (in particular, every n-point subset of L ₁) admits a Fréchet embedding into Euclidean space with distortion , a result which is tight up to the O(log log n) factor, even for Euclidean metrics. This strengthens our recent work on the Euclidean distortion of metrics of negative into Euclidean space. S. Arora supported by David and Lucile Packard Fellowship and NSF grant CCR-0205594. J.R. Lee supported by NSF grant CCR-0121555, NSF 0514993, NSF 0528414 and an NSF Graduate Research Fellowship.     

Extended Lorentz cones and mixed complementarity problems

In this paper we extend the notion of a Lorentz cone in a Euclidean space as follows: we divide the index set corresponding to the coordinates of points in two disjoint classes. By definition a point belongs to an extended Lorentz cone associated with this division, if the coordinates corresponding to one class are at least as large as the norm of the vector formed by the coordinates corresponding to the other class. We call a closed convex set isotone projection set with respect to a pointed closed convex cone if the projection onto the set is isotone (i.e., order preserving) with respect to the partial order defined by the cone. We determine the isotone projection sets with respect to an extended Lorentz cone. In particular, a Cartesian product between an Euclidean space and any closed convex set in another Euclidean space is such a set. We use this property to find solutions of general mixed complementarity problems recursively.     

Ruled surfaces with vanishing second Gaussian curvature

For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid.     

Singularities of Monotone Vector Fields and an Extragradient-type Algorithm

Bearing in mind the notion of monotone vector field on Riemannian manifolds, see [12--16], we study the set of their singularities and for a particularclass of manifolds develop an extragradient-type algorithm convergent to singularities of such vector fields. In particular, our method can be used forsolving nonlinear constrained optimization problems in Euclidean space, with a convex objective function and the constraint set a constant curvature Hadamard manifold. Our paper shows how tools of convex analysis on Riemannian manifolds can be used to solve some nonconvex constrained problem in a Euclidean space.O.P. Ferreira- was supported in part by CAPES, FUNAPE (UFG) and (CNPq).S.Z. Németh- was supported in part by grant No.T029572 of the National Research Foundation of Hungary.     

The three-dimensional Poincaré continued fraction algorithm

It is proved here that for Lebesgue-almost every line in the three-dimensional Euclidean space, the Poincaré continued fraction algorithm fixes a vertex. Besides, the algorithm is nonergodic, although the Gauss map, defined by the algorithm, has an attractor and is ergodic. It is also shown that the Euclidean algorithm and the horocycle flow are orbit equivalent. Partially supported by CNPq(Brazil) grant no. 30.1456-80.     

Hardy and Rellich type inequalities with remainders

Czechoslovak Mathematical Journal - Hardy and Rellich type inequalities with an additional term are proved for compactly supported smooth functions on open subsets of the Euclidean space. We obtain...     

A note on the instability of embeddings of Cauchy-Riemann manifolds

We prove that there are compact strictly pseudoconvex CR manifolds, embedded into some Euclidean space, that admit small deformations that are also embeddable but their embeddings cannot be chosen close to the original embedding. Both authors were partially supported by NSF grants.     

Sobolev classes of Banach space-valued functions and quasiconformal mappings

We give a definition for the class of Sobolev functions from a metric measure space into a Banach space. We give various characterizations of Sobolev classes and study the absolute continuity in measure of Sobolev mappings in the “borderline case”. We show under rather weak assumptions on the source space that quasisymmetric homeomorphisms belong to a Sobolev space of borderline degree; in particular, they are absolutely continuous. This leads to an analytic characterization of quasiconformal mappings between Ahlfors regular Loewner spaces akin to the classical Euclidean situation. As a consequence, we deduce that quasisymmetric maps respect the Cheeger differentials of Lipschitz functions on metric measure spaces with borderline Poincaré inequality. J. H. supported by NSF grant DMS9970427. P. K. supported by the Academy of Finland, project 39788. N. S. supported in part by Enterprise Ireland. J. T. T. supported by an NSF Postdoctoral Research Fellowship.     

A subgaussian embedding theorem

We prove a subgaussian extension of a Gaussian result on embedding subsets of a Euclidean space into normed spaces. Using the concentration of a random subgaussian vector around its mean we obtain an isomorphic (rather than almost isometric) result, under an additional cotype assumption on the normed space considered. Partially supported by an Australian Research Council Discovery grant. This author holds the Canada Research Chair in Geometric Analysis.     

A dual differentiable exact penalty function

A new penalty function is associated with an inequality constrained nonlinear programming problem via its dual. This penalty function is globally differentiable if the functions defining the original problem are twice globally differentiable. In addition, the penalty parameter remains finite. This approach reduces the original problem to a simple problem of maximizing a globally differentiable function on the product space of a Euclidean space and the nonnegative orthant of another Euclidean space. Many efficient algorithms exist for solving this problem. For the case of quadratic programming, the penalty function problem can be solved effectively by successive overrelaxation (SOR) methods which can handle huge problems while preserving sparsity features. Sponsored by the United States Army under Contract No. DAAG 29-80-C-0041. This material is based upon work supported by the National Science Foundation under Grants No. MCS-790166 and ENG-7903881.     

Hardy-type inequalities on planar and spatial open sets

Several Hardy-type inequalities with explicit constants are proved for compactly supported smooth functions on open sets in the Euclidean space ? ⁿ .     

An extension of Takahashi's theorem

A classical result of T. Takahashi [8] is generalized to the case of hypersurfaces in the Euclidean space E ^m . More concretely, we classify Euclidean hypersurfaces whose coordinate functions in E ^m are eigenfunctions of their Laplacian.Partially supported by a CAICYT Grant PR84-1242-C02-02 Spain.     

Approximation properties of a multilayered feedforward artificial neural network

We prove that an artificial neural network with multiple hidden layers and akth-order sigmoidal response function can be used to approximate any continuous function on any compact subset of a Euclidean space so as to achieve the Jackson rate of approximation. Moreover, if the function to be approximated has an analytic extension, then a nearly geometric rate of approximation can be achieved. We also discuss the problem of approximation of a compact subset of a Euclidean space with such networks with a classical sigmoidal response function.Dedicated to Dr. C.A. Micchelli on the occasion of his fiftieth birthday, December 1992Research supported in part by AFOSR Grant No. 226 113 and by the AvH Foundation.     

Euclidean submanifolds with Jacobi mean curvature vector field

We study submanifolds in the Euclidean space whose mean curvature vector field is a Jacobi field. First, we characterize them and produce non-trivial (non-minimal) examples and then, we look for additional conditions which imply minimality.Research partially supported by a DGICYT grant No PB94-0705-C02-01 and by a grant of Gobierno Vasco PI95/95     

On the enumeration of Archimedean polyhedra in the Lobachevsky space

We describe the class of Archimedean polyhedra in the three-dimensional Lobachevsky space, which technically reduces to studying Archimedean tilings of the Lobachevsky plane. We analyze the possibility of obtaining Archimedean tilings by methods that are usually applied on the sphere and in the Euclidean plane. It is pointed out that such tilings can be constructed by using certain types of Fedorov groups in the Lobachevsky plane. We propose a general approach to the problem of classifying Archimedean tilings of the Lobachevsky plane.     

Minimal total absolute curvature for immersions

Immersions or maps of closed manifolds in Euclidean space, of minimal absolute total curvature are called tight in this paper. (They were called convex in [25].) After the definition in Chapter 1, many examples in Chapter 2, and some special topics in Chapter 3, we prove in Chapter 4 that topological tight immersions ofn-spheres are only of the expected type, namely embeddings onto the boundary of a convexn+1-dimensional body. This generalises a theorem of Chern and Lashof in the smooth case. In Chapter 5 we show that many manifolds exist that have no tight smooth immersion in any Euclidean space.This research was partially supported by National Science Foundation grant GP-7952X1.     

Translation surfaces with constant mean curvature in 3-dimensional spaces

We give the classification of the translation surfaces with constant mean curvature or constant Gauss curvature in 3-dimensional Euclidean space E³ and 3-dimensional Minkowski space E ₁ ³ .The author is supported by the EDU. COMM. of CHINA, the NSF of Liaoning and the Northeastern University.Dedicated to Professor Udo Simon on the occation of his sixtieth birthday